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A weak MLMC scheme for L\'evy-copula-driven SDEs with applications to the pricing of credit, equity and interest rate derivatives

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  • Aleksandar Mijatovi'c
  • Romain Palfray

Abstract

This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for L\'evy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain approximation) of the pure-jump component of the driving L\'evy process and is particularly suited if the multidimensional driver is given by a L\'evy copula. The multilevel version of the algorithm requires a new coupling of the approximate L\'evy drivers in the consecutive levels of the scheme, which is defined via a coupling of the corresponding Poisson point processes. The multilevel scheme is weak in the sense that the bound on the level variances is based on the coupling alone without requiring strong convergence. Moreover, the coupling is natural for the proposed discretization of jumps and is easy to simulate. The approximation scheme and its multilevel analogous are applied to examples taken from mathematical finance, including the pricing of credit, equity and interest rate derivatives.

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  • Aleksandar Mijatovi'c & Romain Palfray, 2022. "A weak MLMC scheme for L\'evy-copula-driven SDEs with applications to the pricing of credit, equity and interest rate derivatives," Papers 2211.02528, arXiv.org.
    • Handle: RePEc:arx:papers:2211.02528
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    1. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    2. Zorana Grbac & David Krief & Peter Tankov, 2015. "Approximate Option Pricing in the L\'evy Libor Model," Papers 1511.08466, arXiv.org, revised Jul 2016.
    3. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    4. Paul Glasserman & S. G. Kou, 2003. "The Term Structure of Simple Forward Rates with Jump Risk," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 383-410, July.
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